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Math Help - polynomial equation

  1. #1
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    polynomial equation

    calculate all polynomial p(x) for which (p(x))^2+(p(x-1))^2+1=(P(x)-x)^2
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    Quote Originally Posted by jacks View Post
    calculate all polynomial p(x) for which (p(x))^2+(p(x-1))^2+1=(p(x)-x)^2
    If you put x=0 in this equation then you get (p(-1))^2+1=0. This cannot happen if the polynomial has real coefficients. So there are no such real polynomials. If complex coefficients are allowed, then p(-1)=\pm\sqrt{-1}, but I don't see where to go from there.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jacks View Post
    calculate all polynomial p(x) for which (p(x))^2+(p(x-1))^2+1=(P(x)-x)^2
    If we presume that p(x) is going to have at least real coefficients we can expland the definig equation to get
    \displaystyle p(x+1) = \frac{1}{2}(x + 1) - \frac{p^2(x)}{2(x + 1)}

    The above says that p(x) must contain a factor of x + 1 and since no real p(0) exists (ala Opalg) we must also require that p(x) contains a factor of x as well, so we have
    \displaystyle p(x) = x(x + 1)q(x)

    where q(x) is a polynomial. This expression needs to be run through the original equation to get restrictions on q(x), but I can't make any sense out of the resulting equation.

    -Dan
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